The matrix is constructed exactly like the matrix for the fixed effects, and the matrix is constructed to correspond to the effects constituting . The structures of and are defined by using the TYPE=
option described on . The random effects can be classification or continuous effects, and multiple RANDOM statements are
possible.

Some reserved keywords have special significance in the random-effects list.
You can specify INTERCEPT (or INT) as a random effect to indicate the intercept. PROC GLIMMIX does not include the intercept
in the RANDOM statement by default as it does in the MODEL
statement.
You can specify the _RESIDUAL_ keyword (or RESID, RESIDUAL, _RESID_) before the option slash (/) to indicate a residual-type
(R-side) random component that defines the matrix. Basically, the _RESIDUAL_ keyword takes the place of the random-effect if you want to specify
R-side variances and covariance structures. These keywords take precedence over variables in the data set with the same name. If your data or the covariance structure requires that
an effect is specified, you can use the RESIDUAL
option to instruct the GLIMMIX procedure to model the R-side variances and covariances.

In order to add an overdispersion component to the variance function, simply specify a single residual random component. For
example, the following statements fit a polynomial Poisson regression model with overdispersion. The variance function is replaced by :

You can specify the following options in the RANDOM statement after a slash (/).

ALPHA=number

requests that a t-type confidence interval with confidence
level 1 – number be constructed for the predictors of G-side random effects in this statement. The value of number must be between 0 and 1; the default is 0.05. Specifying the ALPHA= option implies the CL
option.

CL

requests that t-type confidence limits
be constructed for each of the predictors of G-side random effects in this statement. The confidence level is 0.95 by default;
this can be changed with the ALPHA=
option. The CL option implies the SOLUTION
option.

G

requests that the estimated matrix be displayed for G-side
random effects associated with this RANDOM statement. PROC GLIMMIX displays blanks for values that are 0.

GC

displays the lower-triangular Cholesky root of the estimated
matrix for G-side random effects.

GCI

displays the inverse Cholesky root of the estimated matrix
for G-side random effects.

GCOORD=LAST | FIRST | MEAN

determines how the GLIMMIX procedure associates coordinates for TYPE=SP()
covariance structures with effect levels for G-side random effects. In these covariance structures, you specify one or more
variables that identify the coordinates of a data point. The levels of classification variables, on the other hand, can occur
multiple times for a particular subject. For example, in the following statements the same level of A can occur multiple times, and the associated values of x might be different:

The GCOORD=LAST option determines the coordinates for a level of the random effect from the last observation associated with
the level. Similarly, the GCOORD=FIRST and GCOORD=MEAN options determine the coordinate from the first observation and from
the average of the observations. Observations not used in the analysis are not considered in determining the first, last,
or average coordinate. The default is GCOORD=LAST.

GCORR

displays the correlation matrix that corresponds to the estimated matrix for G-side random effects.

GI

displays the inverse of the estimated matrix
for G-side random effects.

GROUP=effect
GRP=effect

identifies groups by which to vary the covariance parameters.
Each new level of the grouping effect produces a new set of covariance parameters. Continuous variables and computed variables
are permitted as group effects. PROC GLIMMIX does not sort by the values of the continuous variable; rather, it considers
the data to be from a new group whenever the value of the continuous variable changes from the previous observation. Using
a continuous variable decreases execution time for models with a large number of groups and also prevents the production of
a large "Class Levels Information" table.

Specifying a GROUP effect can greatly increase the number of estimated covariance parameters, which can adversely affect the
optimization process.

KNOTINFO

displays the number and coordinates of the knots as determined by the KNOTMETHOD=
option.

KNOTMAX=number-list

provides upper limits for the values of random effects used in the
construction of knots for TYPE=RSMOOTH
. The items in number-list correspond to the random effects of the radial smooth. If the KNOTMAX= option is not specified, or if the value associated
with a particular random effect is set to missing, the maximum is based on the values in the data set for KNOTMETHOD=
EQUAL or KNOTMETHOD=
KDTREE, and is based on the values in the knot data set for KNOTMETHOD=
DATA.

determines the method of constructing knots for the radial smoother
fit with the TYPE=RSMOOTH
covariance structure and the TYPE=PSPLINE
covariance structure.

Unless you select the TYPE=RSMOOTH
or TYPE=PSPLINE
covariance structure, the KNOTMETHOD= option has no effect. The default for TYPE=RSMOOTH
is KNOTMETHOD=KDTREE. For TYPE=PSPLINE
, only equally spaced knots are used and you can use the optional numberlist argument of KNOTMETHOD=EQUAL to determine the number of interior knots for TYPE=PSPLINE
.

Knot Construction for TYPE=RSMOOTH

PROC GLIMMIX fits a low-rank smoother, meaning that the number of knots is considerably less than the number of observations.
By default, PROC GLIMMIX determines the knot locations based on the vertices of a k-d tree (Friedman, Bentley, and Finkel 1977; Cleveland and Grosse 1991). The k-d tree is a tree data structure that is useful for efficiently determining the m nearest neighbors of a point. The k-d tree also can be used to obtain a grid of points that adapts to the configuration of the data. The process starts with
a hypercube that encloses the values of the random effects. The space is then partitioned recursively by splitting cells at
the median of the data in the cell for the random effect. The procedure is repeated for all cells that contain more than a
specified number of points, b. The value b is called the bucket size.

The k-d tree is thus a division of the data into cells such that cells representing leaf nodes contain at most b values. You control the building of the k-d tree through the BUCKET= tree-option. You control the construction of knots from the cell coordinates of the tree with the other options as follows.

BUCKET=number

determines the bucket size b. A larger bucket size will result in
fewer knots. For k-d trees in more than one dimension, the correspondence between bucket size and number of knots is difficult to determine.
It depends on the data configuration and on other suboptions. In the multivariate case, you might need to try out different
bucket sizes to obtain the desired number of knots. The default value of number is 4 for univariate trees (a single random effect) and in the multidimensional case.

KNOTTYPE=type

specifies whether the knots are based on vertices of the tree cells
or the centroid. The two possible values of type are VERTEX and CENTER. The default is KNOTTYPE=VERTEX. For multidimensional smoothing, such as smoothing across irregularly
shaped spatial domains, the KNOTTYPE=CENTER option is useful to move knot locations away from the bounding hypercube toward
the convex hull.

NEAREST

specifies that knot coordinates are the coordinates of the
nearest neighbor of either the centroid or vertex of the cell, as determined by the KNOTTYPE= suboption.

TREEINFO

displays details about the construction of the k-d tree, such
as the cell splits and the split values.

See the section Knot Selection for a detailed example of how the specification of the bucket size translates into the construction of a k-d tree and the spline knots.

The KNOTMETHOD=EQUAL option enables you to define a regular grid of knots. By default, PROC GLIMMIX constructs 10 knots for
one-dimensional smooths and 5 knots in each dimension for smoothing in higher dimensions. You can specify a different number
of knots with the optional number-list. Missing values in the number-list are replaced with the default values. A minimum of two knots in each dimension is required. For example, the following statements
use a rectangular grid of 35 knots, five knots for x1 combined with seven knots for x2:

When you use the NOFIT
option in the PROC GLIMMIX
statement, the GLIMMIX procedure computes the knots but does not fit the model. This can be useful if you want to compare
knot selections with different suboptions of KNOTMETHOD=KDTREE. Suppose you want to determine the number of knots based on
a particular bucket size. The following statements compute and display the knots in a bivariate smooth, constructed from nearest
neighbors of the vertices of a k-d tree with bucket size 10:

You can specify a data set that contains variables whose values give the knot coordinates with the KNOTMETHOD=DATA option.
The data set must contain numeric variables with the same name as the radial smoothing random-effects. PROC GLIMMIX uses only the unique knot coordinates in the knot data set. This option is useful to provide knot coordinates
different from those that can be produced from a k-d tree. For example, in spatial problems where the domain is irregularly shaped, you might want to determine knots by a space-filling
algorithm. The following SAS statements invoke the OPTEX procedure to compute 45 knots that uniformly cover the convex hull
of the data locations (see
SAS/QC User's Guide for details about the OPTEX procedure).

Only evenly spaced knots are supported when you fit penalized B-splines with the GLIMMIX procedure. For the TYPE=PSPLINE
covariance structure, the number-list argument specifies the number m of interior knots, the default is . Suppose that and denote the smallest and largest values, respectively. For a B-spline of degree d (De Boor 2001), the interior knots are supplemented with d exterior knots below and exterior knots above . PROC GLIMMIX computes the location of these knots as follows. Let , then interior knots are placed at

The exterior knots are also evenly spaced with step size and start at times the machine epsilon. At least one interior knot is required.

KNOTMIN=number-list

provides lower limits for the values of random effects used in the
construction of knots for TYPE=RSMOOTH
. The items in number-list correspond to the random effects of the radial smooth. If the KNOTMIN= option is not specified, or if the value associated
with a particular random effect is set to missing, the minimum is based on the values in the data set for KNOTMETHOD=
EQUAL or KNOTMETHOD=
KDTREE, and is based on the values in the knot data set for KNOTMETHOD=
DATA.

LDATA=SAS-data-set

reads the coefficient matrices
for the TYPE=LIN(q)
option. You can specify the LDATA= data set in a sparse or dense form. In the sparse form the data set must contain the numeric
variables Parm, Row, Col, and Value. The Parm variable contains the indices of the matrices. The Row and Col variables identify the position within a matrix and the Value variable contains the matrix element. Values not specified for a particular row and column are set to zero. Missing values
are allowed in the Value column of the LDATA= data set; these values are also replaced by zeros. The sparse form is particularly useful if the matrices have only a few nonzero elements.

In the dense form the LDATA= data set contains the numeric variables Parm and Row (with the same function as above), in addition to the numeric variables Col1–Colq. If you omit one or more of the Col1–Colq variables from the data set, zeros are assumed for the respective rows and columns of the matrix. Missing values for Col1–Colq are ignored in the dense form.

The GLIMMIX procedure assumes that the matrices are symmetric. In the sparse LDATA= form you do not need to specify off-diagonal elements in position and . One of them is sufficient. Row-column indices are converted in both storage forms into positions in lower triangular storage.
If you specify multiple values in row and column of a particular matrix, only the last value is used. For example, assume you are specifying elements of a matrix. The lower triangular storage of matrix defined by

data ldata;
input parm row col value;
datalines;
3 2 1 2
3 1 2 5
;

is

NOFULLZ

eliminates the columns in corresponding to missing levels of
random effects involving CLASS variables. By default, these columns are included in . It is sufficient to specify the NOFULLZ option on any G-side RANDOM statement.

RESIDUAL
RSIDE

specifies that the random effects listed in this statement
be R-side effects. You use the RESIDUAL option in the RANDOM statement if the nature of the covariance structure requires
you to specify an effect. For example, if it is necessary to order the columns of the R-side AR(1) covariance structure by
the time variable, you can use the RESIDUAL option as in the following statements:

class time id;
random time / subject=id type=ar(1) residual;

SOLUTION
S

requests that the solution for the
random-effects parameters be produced, if the statement defines G-side random effects.

The numbers displayed in the Std Err Pred column of the "Solution for Random Effects" table are not the standard errors of
the displayed in the Estimate column; rather, they are the square roots of the prediction errors , where is the predictor of the ith random effect and is the ith random effect. In pseudo-likelihood methods that are based on linearization, these EBLUPs are the estimated best linear
unbiased predictors in the linear mixed pseudo-model. In models fit by maximum likelihood by using the Laplace approximation
or by using adaptive quadrature, the SOLUTION option displays the empirical Bayes estimates (EBE) of .

SUBJECT=effect
SUB=effect

identifies the subjects in your generalized linear mixed model.
Complete independence is assumed across subjects. Specifying a subject effect is equivalent to nesting all other effects in
the RANDOM statement within the subject effect.

Continuous variables and computed variables are permitted with the SUBJECT=
option. PROC GLIMMIX does not sort by the values of the continuous variable but considers the data to be from a new subject
whenever the value of the continuous variable changes from the previous observation. Using a continuous variable can decrease
execution time for models with a large number of subjects and also prevents the production of a large "Class Levels Information"
table.

TYPE=covariance-structure

specifies the covariance structure of for G-side effects and
the covariance structure of for R-side effects.

Although a variety of structures are available, many applications call for either simple diagonal (TYPE=VC
) or unstructured covariance matrices. The TYPE=VC
(variance components) option is the default structure, and it models a different variance component for each random effect.
It is recommended to model unstructured covariance matrices in terms of their Cholesky parameterization (TYPE=CHOL
) rather than TYPE=UN
.

If you want different covariance structures in different parts of , you must use multiple RANDOM statements with different TYPE= options.

Valid values for covariance-structure are as follows. Examples are shown in Table 45.19.

The variances and covariances in the formulas that follow in the TYPE= descriptions are expressed in terms of generic random
variables and . They represent the G-side random effects or the residual random variables for which the or matrices are constructed.

ANTE(1)

specifies a first-order ante-dependence structure
(Kenward 1987; Patel 1991) parameterized in terms of variances and correlation parameters. If t ordered random variables have a first-order ante-dependence structure, then each , , is independent of all other , given . This Markovian structure is characterized by its inverse variance matrix, which is tridiagonal. Parameterizing an ANTE(1)
structure for a random vector of size t requires 2t – 1 parameters: variances and t – 1 correlation parameters . The covariances among random variables and are then constructed as

The values and are derived for the ith and jth observations, respectively, and are not necessarily the observation numbers. For example, in the following statements the
values correspond to the class levels for the time effect of the ith and jth observation within a particular subject:

where and . PROC GLIMMIX imposes the constraints and for stationarity, although for some values of and in this region the resulting covariance matrix is not positive definite. When the estimated value of becomes negative, the computed covariance is multiplied by to account for the negativity.

CHOL<(q)>

specifies an unstructured variance-covariance matrix
parameterized through its Cholesky root. This parameterization ensures that the resulting variance-covariance matrix is at
least positive semidefinite. If all diagonal values are nonzero, it is positive definite. For example, a unstructured covariance matrix can be written as

Without imposing constraints on the three parameters, there is no guarantee that the estimated variance matrix is positive
definite. Even if and are nonzero, a large value for can lead to a negative eigenvalue of . The Cholesky root of a positive definite matrix is a lower triangular matrix such that . The Cholesky root of the above matrix can be written as

The elements of the unstructured variance matrix are then simply , , and . Similar operations yield the generalization to covariance matrices of higher orders.

For example, the following statements model the covariance matrix of each subject as an unstructured matrix:

The GLIMMIX procedure constrains the diagonal elements of the Cholesky root to be positive. This guarantees a unique solution
when the matrix is positive definite.

The optional order parameter determines how many bands below the diagonal are modeled. Elements in the lower triangular portion of in bands higher than q are set to zero. If you consider the resulting covariance matrix , then the order parameter has the effect of zeroing all off-diagonal elements that are at least q positions away from the diagonal.

Because of its good computational and statistical properties, the Cholesky root parameterization is generally recommended
over a completely unstructured covariance matrix (TYPE=UN
). However, it is computationally slightly more involved.

CS

specifies the compound-symmetry structure, which
has constant variance and constant covariance

The compound symmetry structure arises naturally with nested random effects, such as when subsampling error is nested within
experimental error. The models constructed with the following two sets of GLIMMIX statements have the same marginal variance
matrix, provided is positive:

In the first case, the block*A random effect models the G-side experimental error. Because the distribution defaults to the normal, the matrix is of form (see Table 45.20), and is the subsampling error variance. The marginal variance for the data from a particular experimental unit is thus . This matrix is of compound symmetric form.

Hierarchical random assignments or selections, such as subsampling or split-plot designs, give rise to compound symmetric
covariance structures. This implies exchangeability of the observations on the subunit, leading to constant correlations between
the observations. Compound symmetric structures are thus usually not appropriate for processes where correlations decline
according to some metric, such as spatial and temporal processes.

Note that R-side compound-symmetry structures do not impose any constraint on . You can thus use an R-side TYPE=CS structure to emulate a variance-component model with unbounded estimate of the variance
component.

CSH

specifies the heterogeneous compound-symmetry structure, which is an
equi-correlation structure but allows for different variances

FA(q)

specifies the factor-analytic structure with q factors
(Jennrich and Schluchter 1986). This structure is of the form , where is a rectangular matrix and is a diagonal matrix with t different parameters. When , the elements of in its upper-right corner (that is, the elements in the ith row and jth column for ) are set to zero to fix the rotation of the structure.

FA0(q)

specifies a factor-analytic structure with q factors of the
form , where is a rectangular matrix and t is the dimension of . When , is a lower triangular matrix. When —that is, when the number of factors is less than the dimension of the matrix—this structure is nonnegative definite but not
of full rank. In this situation, you can use it to approximate an unstructured covariance matrix.

HF

specifies a covariance structure that satisfies the general
Huynh-Feldt condition (Huynh and Feldt 1970). For a random vector with t elements, this structure has positive parameters and covariances

A covariance matrix generally satisfies the Huynh-Feldt condition if it can be written as . The preceding parameterization chooses . Several simpler covariance structures give rise to covariance matrices that also satisfy the Huynh-Feldt condition. For
example, TYPE=CS
, TYPE=VC
, and TYPE=UN(1)
are nested within TYPE=HF. You can use the COVTEST
statement to test the HF structure against one of these simpler structures. Note also that the HF structure is nested within
an unstructured covariance matrix.

The TYPE=HF covariance structure can be sensitive to the choice of starting values and the default MIVQUE(0) starting values
can be poor for this structure; you can supply your own starting values with the PARMS
statement.

LIN(q)

specifies a general linear covariance structure with q parameters.
This structure consists of a linear combination of known matrices that you input with the LDATA=
option. Suppose that you want to model the covariance of a random vector of length t, and further suppose that are symmetric ) matrices constructed from the information in the LDATA=
data set. Then,

where denotes the element in row i, column j of matrix .

Linear structures are very flexible and general. You need to exercise caution to ensure that the variance matrix is positive
definite. Note that PROC GLIMMIX does not impose boundary constraints on the parameters of a general linear covariance structure. For example, if classification variable A has 6 levels, the following statements fit a variance component structure for the random effect without boundary constraints:

requests that PROC GLIMMIX form a B-spline basis and fits a
penalized B-spline (P-spline, Eilers and Marx 1996) with random spline coefficients. This covariance structure is available only for G-side random effects and only a single
continuous random effect can be specified with TYPE=PSPLINE. As for TYPE=RSMOOTH, PROC GLIMMIX forms a modified matrix and fits a mixed model in which the random variables associated with the columns of are independent with a common variance. The matrix is constructed as follows.

Denote as the matrix of B-splines of degree d and denote as the matrix of rth-order differences. For example, for K = 5,

Then, the matrix used in fitting the mixed model is the matrix

The construction of the B-spline knots is controlled with the KNOTMETHOD=
EQUAL(m) option and the DEGREE=d suboption of TYPE=PSPLINE. The total number of knots equals the number m of equally spaced interior knots plus d knots at the low end and knots at the high end. The number of columns in the B-spline basis equals K = m + d + 1. By default, the interior knots exclude the minimum and maximum of the random-effect values and are based on m – 1 equally spaced intervals. Suppose and are the smallest and largest random-effect values; then interior knots are placed at

Details about the computation and properties of B-splines can be found in De Boor (2001). You can extend or limit the range of the knots with the KNOTMIN=
and KNOTMAX=
options. Table 45.18 lists some of the parameters that control this covariance type and their relationships.

Table 45.18: P-Spline Parameters

Parameter

Description

d

Degree of B-spline, default d = 3

r

Order of differencing in construction of , default r = 3

m

Number of interior knots, default

Total number of knots

Number of columns in B-spline basis

Number of columns in

You can specify the following options for TYPE=PSPLINE:

DEGREE=d

specifies the degree of the B-spline. The default is d = 3.

DIFFORDER=r

specifies the order of the differencing matrix . The default and maximum is r = 3.

RSMOOTH<(m | NOLOG)>

specifies a radial smoother covariance structure for G-side random
effects. This results in an approximate low-rank thin-plate spline where the smoothing parameter is obtained by the estimation
method selected with the METHOD=
option of the PROC GLIMMIX
statement. The smoother is based on the automatic smoother in Ruppert, Wand, and Carroll (2003, Chapter 13.4–13.5), but with a different method of selecting the spline knots. See the section Radial Smoothing Based on Mixed Models for further details about the construction of the smoother and the knot selection.

Radial smoothing is possible in one or more dimensions. A univariate smoother is obtained with a single random effect, while
multiple random effects in a RANDOM statement yield a multivariate smoother. Only continuous random effects are permitted
with this covariance structure. If denotes the number of continuous random effects in the RANDOM statement, then the covariance structure of the random effects
is determined as follows. Suppose that denotes the vector of random effects for the ith observation. Let denote the vector of knot coordinates, , and K is the total number of knots. The Euclidean distance between the knots is computed as

and the distance between knots and effects is computed as

The matrix for the GLMM is constructed as

where the matrix has typical element

and the matrix has typical element

The exponent in these expressions equals , where the optional value m corresponds to the derivative penalized in the thin-plate spline. A larger value of m will yield a smoother fit. The GLIMMIX procedure requires p > 0 and chooses by default m = 2 if and otherwise. The NOLOG option removes the and terms from the computation of the and matrices when is even; this yields invariance under rescaling of the coordinates.

Finally, the components of are assumed to have equal variance . The "smoothing parameter" of the low-rank spline is related to the variance components in the model, . See Ruppert, Wand, and Carroll (2003) for details. If the conditional distribution does not provide a scale parameter , you can add a single R-side residual parameter.

The knot selection is controlled with the KNOTMETHOD=
option. The GLIMMIX procedure selects knots automatically based on the vertices of a k-d tree or reads knots from a data set that you supply. See the section Radial Smoothing Based on Mixed Models for further details on radial smoothing in the GLIMMIX procedure and its connection to a mixed model formulation.

SIMPLE

is an alias for TYPE=VC.

SP(EXP)(c-list)

models an exponential spatial or temporal covariance structure,
where the covariance between two observations depends on a distance metric . The c-list contains the names of the numeric variables used as coordinates to determine distance. For a stochastic process in , there are k elements in c-list. If the vectors of coordinates for observations i and j are and , then PROC GLIMMIX computes the Euclidean distance

The covariance between two observations is then

The parameter is not what is commonly referred to as the range parameter in geostatistical applications. The practical range of a (second-order
stationary) spatial process is the distance at which the correlations fall below 0.05. For the SP(EXP) structure, this distance is . PROC GLIMMIX constrains to be positive.

SP(GAU)(c-list)

models a Gaussian covariance structure,

See TYPE=SP(EXP) for the computation of the distance . The parameter is related to the range of the process as follows. If the practical range is defined as the distance at which the correlations fall below 0.05, then . PROC GLIMMIX constrains to be positive. See TYPE=SP(EXP) for the computation of the distance from the variables specified in c-list.

SP(MAT)(c-list)

models a covariance structure in the Matérn class of covariance
functions (Matérn 1986). The covariance is expressed in the parameterization of Handcock and Stein (1993); Handcock and Wallis (1994); it can be written as

The function is the modified Bessel function of the second kind of (real) order . The smoothness (continuity) of a stochastic process with covariance function in the Matérn class increases with . This class thus enables data-driven estimation of the smoothness properties of the process. The covariance is identical
to the exponential model for (TYPE=SP(EXP)(c-list)), while for the model advocated by Whittle (1954) results. As , the model approaches the Gaussian covariance structure (TYPE=SP(GAU)(c-list)).

Note that the MIXED procedure offers covariance structures in the Matérn class in two parameterizations, TYPE=SP(MATERN) and
TYPE=SP(MATHSW). The TYPE=SP(MAT) in the GLIMMIX procedure is equivalent to TYPE=SP(MATHSW) in the MIXED procedure.

Computation of the function and its derivatives is numerically demanding; fitting models with Matérn covariance structures can be time-consuming. Good
starting values are essential.

SP(POW)(c-list)

models a power covariance structure,

where . This is a reparameterization of the exponential structure, TYPE=SP(EXP). Specifically, . See TYPE=SP(EXP) for the computation of the distance from the variables specified in c-list. When the estimated value of becomes negative, the computed covariance is multiplied by to account for the negativity.

SP(POWA)(c-list)

models an anisotropic power covariance structure in k dimensions,
provided that the coordinate list c-list has k elements. If denotes the coordinate for the ith observation of the mth variable in c-list, the covariance between two observations is given by

Note that for k = 1, TYPE=SP(POWA) is equivalent to TYPE=SP(POW), which is itself a reparameterization of TYPE=SP(EXP). When the estimated
value of becomes negative, the computed covariance is multiplied by to account for the negativity.

SP(SPH)(c-list)

models a spherical covariance structure,

The spherical covariance structure has a true range parameter. The covariances between observations are exactly zero when
their distance exceeds . See TYPE=SP(EXP) for the computation of the distance from the variables specified in c-list.

TOEP

models a Toeplitz covariance structure. This structure can be viewed
as an autoregressive structure with order equal to the dimension of the matrix,

TOEP(q)

specifies a banded Toeplitz structure,

This can be viewed as a moving-average structure with order equal to q – 1. The specification TYPE=TOEP(1) is the same as , and it can be useful for specifying the same variance component for several effects.

TOEPH<(q)>

models a Toeplitz covariance structure. The correlations of this
structure are banded as the TOEP or TOEP(q) structures, but the variances are allowed to vary:

The correlation parameters satisfy . If you specify the optional value q, the correlation parameters with are set to zero, creating a banded correlation structure. The specification TYPE=TOEPH(1) results in a diagonal covariance
matrix with heterogeneous variances.

UN<(q)>

specifies a completely general (unstructured) covariance matrix
parameterized directly in terms of variances and covariances,

The variances are constrained to be nonnegative, and the covariances are unconstrained. This structure is not constrained
to be nonnegative definite in order to avoid nonlinear constraints; however, you can use the TYPE=CHOL structure if you want
this constraint to be imposed by a Cholesky factorization. If you specify the order parameter q, then PROC GLIMMIX estimates only the first q bands of the matrix, setting elements in all higher bands equal to 0.

UNR<(q)>

specifies a completely general (unstructured) covariance matrix
parameterized in terms of variances and correlations,

where denotes the standard deviation and the correlation is zero when and when , provided the order parameter q is given. This structure fits the same model as the TYPE=UN(q) option, but with a different parameterization. The ith variance parameter is . The parameter is the correlation between the ith and jth measurements; it satisfies . If you specify the order parameter q, then PROC GLIMMIX estimates only the first q bands of the matrix, setting all higher bands equal to zero.

VC

specifies standard variance components and is the default structure
for both G-side and R-side covariance structures. In a G-side covariance structure, a distinct variance component is assigned
to each effect. In an R-side structure TYPE=VC is usually used only to add overdispersion effects or with the GROUP=
option to specify a heterogeneous variance model.

requests that blocks of the estimated marginal variance-covariance matrix be displayed in generalized linear mixed models. This matrix is based on the last linearization as described in the section
The Pseudo-model. You can use the value-list to select the subjects for which the matrix is displayed. If value-list is not specified, the matrix for the first subject is chosen.

Note that the value-list refers to subjects as the processing units in the "Dimensions" table. For example, the following statements request that
the estimated marginal variance matrix for the second subject be displayed:

The subject effect for processing in this case is the A effect, because it is contained in the A*B interaction. If there is only a single subject as per the "Dimensions" table, then the V option displays an matrix.

See the section Processing by Subjects for how the GLIMMIX procedure determines the number of subjects in the "Dimensions" table.

The GLIMMIX procedure displays blanks for values that are 0.

VC<=value-list>

displays the lower-triangular Cholesky root of the blocks of the
estimated matrix. See the V
option for the specification of value-list.

VCI<=value-list>

displays the inverse Cholesky root of the blocks of the estimated matrix. See the V
option for the specification of value-list.

VCORR<=value-list>

displays the correlation matrix corresponding to the blocks of the
estimated matrix. See the V
option for the specification of value-list.

VI<=value-list>

displays the inverse of the blocks of the estimated matrix. See the V
option for the specification of value-list.

WEIGHT<=variable>
WT<=variable>

specifies a variable to be used as the weight
for the units at the current level in a weighted multilevel model. If a weight variable is not specified in the WEIGHT option,
a weight of 1 is used. For details on the use of weights in multilevel models, see the section Pseudo-likelihood Estimation for Weighted Multilevel Models.